Problem: Simplify the following expression: $ a = \dfrac{x + 1}{-9x + 7} - \dfrac{-7}{2} $
Explanation: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{2}{2}$ $ \dfrac{x + 1}{-9x + 7} \times \dfrac{2}{2} = \dfrac{2x + 2}{-18x + 14} $ Multiply the second expression by $\dfrac{-9x + 7}{-9x + 7}$ $ \dfrac{-7}{2} \times \dfrac{-9x + 7}{-9x + 7} = \dfrac{63x - 49}{-18x + 14} $ Therefore $ a = \dfrac{2x + 2}{-18x + 14} - \dfrac{63x - 49}{-18x + 14} $ Now the expressions have the same denominator we can simply subtract the numerators: $a = \dfrac{2x + 2 - (63x - 49) }{-18x + 14} $ Distribute the negative sign: $a = \dfrac{2x + 2 - 63x + 49}{-18x + 14}$ $a = \dfrac{-61x + 51}{-18x + 14}$ Simplify the expression by dividing the numerator and denominator by -1: $a = \dfrac{61x - 51}{18x - 14}$